Related papers: The $r$-equilibrium Problem
We consider the dynamics of a collection of particles that interact pairwise and are restricted to move along the real line. Moreover, we focus on the situation in which particles undergo perfectly inelastic collisions when they collide.…
We discuss renormalization of the non-relativistic three-body problem with short-range forces. The problem becomes non-perturbative at momenta of the order of the inverse of the two-body scattering length, and an infinite number of graphs…
It is shown that a class of approximate resonance solutions in the three-body problem under the Newtonian gravitational force are equivalent to quantized solutions of a modified Schr\"odinger equation for a wide range of masses that…
According to Newton's law of gravitation the force between two particles depends upon their inertial, as well as their active and passive gravitational masses. For ordinary matter all three of these are equal and positive. We consider here…
The classic image problem in electromagnetism involves a grounded infinite conducting plane and a point charge. The force of attraction between the point charge and the plane is identified using an equivalent-field picture of an image…
In the recent papers~[18],~[5], respectively, the existence of motions where the perihelions afford periodic oscillations about certain equilibria and the onset of a topological horseshoe have been proved. Such results have been obtained…
By requiring the linear differential operator in Newton's law of motion to be self adjoint, we obtain the field equation for the linear theory, which is the classical electrodynamics. In the process, we are also led to a fundamental…
In this paper, we consider the elliptic relative equilibria of the restricted $N$-body problems, where the $N-1$ primaries form an Euler-Moulton collinear central configuration or a $(1+n)$-gon central configuration. We obtain the…
The relativistic three-body problem is approached via the extension of the SL(2,C) group to the Sp(4,C) one. In terms of Sp(4,C) spinors, a Dirac-like equation with three-body kinematics is composed. After introducing the linear in…
We study the relationship between the solvability of the $L^p$ Dirichlet problem $(D)_p$ and that of the $L^q$ regularity problem $(R)_q$ for second order elliptic equations with bounded measurable coefficients. It is known that the…
An elliptic relative equilibrium (ERE) is a special solution of the planar $N$-body problem generated by a central configuration. Its linear stability depends on the eccentricity $e$ and the masses of the bodies. However, for $e>0$, the…
The role of the SO(2,1) symmetry in General Relativity is analyzed. Cosmological solutions of Einstein field equations invariant with respect to a space-like Lie algebra G_r, with r between 3 and 6 and containing so(2,1) as a subalgebra,…
Consider the dynamics of two point masses on a surface of constant curvature subject to an attractive force analogue of Newton's inverse square law. When the distance between the bodies is sufficiently small, the reduced equations of motion…
We consider the motion of point masses given by a natural extension of Newtonian gravitation to spaces of constant positive curvature. Our goal is to explore the spectral stability of tetrahedral orbits of the corresponding 4-body problem…
Using the relativistic quantum Hamilton-Jacobi equation within the framework of the equivalence postulate, and grounding one self on both relativistic and quantum Lagrangians, we construct a Lagrangian of relativistic quantum system in one…
We generalize the Newtonian n-body problem to spaces of curvature k=constant, and study the motion in the 2-dimensional case. For k>0, the equations of motion encounter non-collision singularities, which occur when two bodies are antipodal.…
The first part of this paper is devoted to an analysis of moment problems in R^n with supports contained in a closed set defined by finitely many polynomial inequalities. The second part of the paper uses the representation results of…
In this paper, we consider the elliptic relative equilibria of the restricted $4$-body problems, where the three primaries form an Euler collinear configuration and the four bodies span $\mathbf{R}^2$. We obtain the symplectic reduction to…
We study the out-of-equilibrium evolution of an O(2)-invariant scalar field in which a conserved charge is stored. We apply a loop expansion of the 2-particle irreducible effective action to 3-loop order. Equations of motion are derived…
We prove that the fixed points of the curved 3-body problem and their associated relative equilibria are Lyapunov stable if the solutions are restricted to $\mathbb S^1$, but unstable if the bodies are considered in $\mathbb S^2$.